cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A293475 a(n) = (3*n + 4)*Pochhammer(n, 4) / 4!.

Original entry on oeis.org

0, 7, 50, 195, 560, 1330, 2772, 5250, 9240, 15345, 24310, 37037, 54600, 78260, 109480, 149940, 201552, 266475, 347130, 446215, 566720, 711942, 885500, 1091350, 1333800, 1617525, 1947582, 2329425, 2768920, 3272360, 3846480, 4498472, 5236000, 6067215, 7000770
Offset: 0

Views

Author

Peter Luschny, Oct 20 2017

Keywords

Links

Crossrefs

Cf. A265609, A293476, A293608, A293615.

Programs

  • Magma
    [0] cat [(3*n + 4)*Factorial(n+3)/(Factorial(n-1)*Factorial(4)): n in [1..30]]; // G. C. Greubel, Nov 20 2017
  • Maple
    A293475 := n -> (3*n + 4)*pochhammer(n, 4)/4!:
seq(A293475(n), n=0..32);
  • Mathematica
    LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 7, 50, 195, 560, 1330}, 32]
Table[n*StirlingS2[n+3, n+1], {n, 0, 50}] (* G. C. Greubel, Nov 20 2017 *)
f[n_] := (3n + 4) Pochhammer[n, 4]/4!; Array[f, 35, 0] (* or *)
CoefficientList[ Series[ x (7 + 8x)/(1 - x)^6, {x, 0, 34}], x] (* Robert G. Wilson v, Nov 21 2017 *)
  • PARI
    for(n=0,30, print1(n*Stirling(n+3, n+1, 2), ", ")) \\ G. C. Greubel, Nov 20 2017
  • PARI
    concat(0, Vec(x*(7 + 8*x) / (1 - x)^6 + O(x^40))) \\ Colin Barker, Nov 21 2017

Formula

a(n) = n*Stirling2(3 + n, 1 + n).
-a(-n-3) = (n + 3)*abs(Stirling1(n+2, n)) for n >= 0.
-a(-n-3) = a(n) + binomial(n+3, 4) for n >= 0.
From Colin Barker, Nov 21 2017: (Start)
G.f.: x*(7 + 8*x) / (1 - x)^6.
a(n) = n*(24 + 62*n + 57*n^2 + 22*n^3 + 3*n^4)/24.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>5. (End)
From Amiram Eldar, Aug 31 2025: (Start)
Sum_{n>=1} 1/a(n) = 27*sqrt(3)*Pi/10 + 243*log(3)/10 - 2473/60.
Sum_{n>=1} (-1)^(n+1)/a(n) = 27*sqrt(3)*Pi/5 + 112*log(2)/5 - 2687/60. (End)

A293608 a(n) = (3*n + 7)*Pochhammer(n, 5) / 4!.

Original entry on oeis.org

0, 50, 390, 1680, 5320, 13860, 31500, 64680, 122760, 218790, 370370, 600600, 939120, 1423240, 2099160, 3023280, 4263600, 5901210, 8031870, 10767680, 14238840, 18595500, 24009700, 30677400, 38820600, 48689550, 60565050, 74760840, 91626080, 111547920, 134954160
Offset: 0

Views

Author

Peter Luschny, Oct 20 2017

Keywords

Links

Crossrefs

Cf. A265609, A293475, A293476, A293615.

Programs

  • Magma
    [0] cat [(3*n + 7)*Factorial(n+4)/(Factorial(4)*Factorial(n-1)): n in [1..30]]; // G. C. Greubel, Nov 20 2017
  • Maple
    A293608 := n -> (3*n+7)*pochhammer(n, 5)/4!:
seq(A293608(n), n=0..11);
  • Mathematica
    LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {0, 50, 390, 1680, 5320, 13860, 31500}, 32]
Table[n*(n+1)*StirlingS2[4 + n, 2 + n], {n,0,50}] (* G. C. Greubel, Nov 20 2017 *)
f[n_] := (3 n + 7) Pochhammer[n, 5]/4!; Array[f, 31, 0] (* or *)
CoefficientList[ Series[10x (5 + 4x)/(1 - x)^7, {x, 0, 30}], x] (* Robert G. Wilson v, Nov 21 2017 *)
  • PARI
    for(n=0,30, print1(n*(n+1)*stirling(4 + n, 2 + n, 2), ", ")) \\ G. C. Greubel, Nov 20 2017
  • PARI
    concat(0, Vec(10*x*(5 + 4*x) / (1 - x)^7 + O(x^40))) \\ Colin Barker, Nov 21 2017

Formula

a(n) = n*(n+1)*Stirling2(4 + n, 2 + n).
-a(-n-4) = a(n) - 30*binomial(n+4, 5)*(n + 2) for n >= 0.
From Colin Barker, Nov 21 2017: (Start)
G.f.: 10*x*(5 + 4*x) / (1 - x)^7.
a(n) = (1/24)*(n*(168 + 422*n + 395*n^2 + 175*n^3 + 37*n^4 + 3*n^5)).
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>6. (End)
From Amiram Eldar, Aug 31 2025: (Start)
Sum_{n>=1} 1/a(n) = 8703/490 - 81*sqrt(3)*Pi/70 - 729*log(3)/70.
Sum_{n>=1} (-1)^(n+1)/a(n) = 81*sqrt(3)*Pi/35 + 416*log(2)/35 - 15298/735. (End)

A293615 a(n) = Pochhammer(n, 5) / 2.

Original entry on oeis.org

0, 60, 360, 1260, 3360, 7560, 15120, 27720, 47520, 77220, 120120, 180180, 262080, 371280, 514080, 697680, 930240, 1220940, 1580040, 2018940, 2550240, 3187800, 3946800, 4843800, 5896800, 7125300, 8550360, 10194660, 12082560, 14240160, 16695360, 19477920, 22619520
Offset: 0

Views

Author

Peter Luschny, Oct 20 2017

Keywords

Links

Crossrefs

Cf. A000389, A265609, A293475, A293476, A293608.

Programs

  • Magma
    [0] cat [Factorial(n+4)/(2*Factorial(n-1)): n in [1..30]]; // G. C. Greubel, Nov 20 2017
  • Maple
    A293615 := n -> pochhammer(n, 5)/2:
seq(A293615(n), n=0..11);
  • Mathematica
    LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 60, 360, 1260, 3360, 7560}, 32]
Table[Pochhammer[n, 5]/2, {n,0,50}] (* G. C. Greubel, Nov 20 2017 *)
  • PARI
    for(n=0,30, print1(n*(n+1)*(n+2)*stirling(4 + n, 3 + n, 2), ", ")) \\ G. C. Greubel, Nov 20 2017
  • PARI
    concat(0, Vec(60*x / (1 - x)^6 + O(x^40))) \\ Colin Barker, Nov 21 2017

Formula

a(n) = n*(n+1)*(n+2)*Stirling2(4 + n, 3 + n).
-a(-n-4) = a(n) for n >= 0.
a(n) = 60*A000389(n+4). - G. C. Greubel, Nov 20 2017
From Colin Barker, Nov 21 2017: (Start)
G.f.: 60*x / (1 - x)^6.
a(n) = (1/2)*(n*(1 + n)*(2 + n)*(3 + n)*(4 + n)).
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>5. (End)
From Amiram Eldar, Aug 31 2025: (Start)
Sum_{n>=1} 1/a(n) = 1/48.
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/3 - 131/144. (End)

A293617 Array of triangles read by ascending antidiagonals, T(m, n, k) = Pochhammer(m, k) * Stirling2(n + m, k + m) with m >= 0, n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 2, 1, 0, 1, 10, 3, 7, 3, 0, 1, 15, 4, 25, 12, 2, 0, 1, 21, 5, 65, 30, 6, 1, 0, 1, 28, 6, 140, 60, 12, 15, 7, 0, 1, 36, 7, 266, 105, 20, 90, 50, 12, 0, 1, 45, 8, 462, 168, 30, 350, 195, 60, 6, 0, 1, 55, 9, 750, 252, 42, 1050, 560, 180, 24, 1, 0
Offset: 0

Views

Author

Peter Luschny, Oct 20 2017

Keywords

Examples

			Array starts:
m\j| 0   1  2     3       4       5       6       7       8       9      10
---|-----------------------------------------------------------------------
m=0| 1,  0, 0,    0,      0,      0,      0,      0,      0,      0,      0
m=1| 1,  1, 1,    1,      3,      2,      1,      7,     12,      6,      1
m=2| 1,  3, 2,    7,     12,      6,     15,     50,     60,     24,     31
m=3| 1,  6, 3,   25,     30,     12,     90,    195,    180,     60,    301
m=4| 1, 10, 4,   65,     60,     20,    350,    560,    420,    120,   1701
m=5| 1, 15, 5,  140,    105,     30,   1050,   1330,    840,    210,   6951
m=6| 1, 21, 6,  266,    168,     42,   2646,   2772,   1512,    336,  22827
m=7| 1, 28, 7,  462,    252,     56,   5880,   5250,   2520,    504,  63987
m=8| 1, 36, 8,  750,    360,     72,  11880,   9240,   3960,    720, 159027
m=9| 1, 45, 9, 1155,    495,     90,  22275,  15345,   5940,    990, 359502
   A000217, A001296,A027480,A002378,A001297,A293475,A033486,A007531,A001298
.
m\j| ...      11      12      13      14
---|-----------------------------------------
m=0| ...,      0,      0,      0,      0, ... [A000007]
m=1| ...,     15,     50,     60,     24, ... [A028246]
m=2| ...,    180,    390,    360,    120, ... [A053440]
m=3| ...,   1050,   1680,   1260,    360, ... [A294032]
m=4| ...,   4200,   5320,   3360,    840, ...
m=5| ...,  13230,  13860,   7560,   1680, ...
m=6| ...,  35280,  31500,  15120,   3024, ...
m=7| ...,  83160,  64680,  27720,   5040, ...
m=8| ..., 178200, 122760,  47520,   7920, ...
m=9| ..., 353925, 218790,  77220,  11880, ...
         A293476,A293608,A293615,A052762, ...
.
The parameter m runs over the triangles and j indexes the triangles by reading them by rows. Let T(m, n) denote the row [T(m, n, k) for 0 <= k <= n] and T(m) denote the triangle [T(m, n) for n >= 0]. Then for instance T(2) is the triangle A053440, T(3, 2) is row 2 of A294032 (which is [25, 30, 12]) and T(3, 2, 1) = 30.
.
Remark: To adapt the sequences A028246 and A053440 to our enumeration use the exponential generating functions exp(x)/(1 - y*(exp(x) - 1)) and exp(x)*(2*exp(x) - y*exp(2*x) + 2*y*exp(x) - 1 - y)/(1 - y*(exp(x) - 1))^2 instead of those indicated in their respective entries.

Links

Crossrefs

A000217(n) = T(n, 1, 0), A001296(n) = T(n, 2, 0), A027480(n) = T(n, 2, 1),
A002378(n) = T(n, 2, 2), A001297(n) = T(n, 3, 0), A293475(n) = T(n, 3, 1),
A033486(n) = T(n, 3, 2), A007531(n) = T(n, 3, 3), A001298(n) = T(n, 4, 0),
A293476(n) = T(n, 4, 1), A293608(n) = T(n, 4, 2), A293615(n) = T(n, 4, 3),
A052762(n) = T(n, 4, 4), A052787(n) = T(n, 5, 5), A000225(n) = T(1, n, 1),
A028243(n) = T(1, n, 2), A028244(n) = T(1, n, 3), A028245(n) = T(1, n, 4),
A032180(n) = T(1, n, 5), A228909(n) = T(1, n, 6), A228910(n) = T(1, n, 7),
A000225(n) = T(2, n, 0), A007820(n) = T(n, n, 0).
A028246(n,k) = T(1, n, k), A053440(n,k) = T(2, n, k), A294032(n,k) = T(3, n, k),
A293926(n,k) = T(n, n, k), A124320(n,k) = T(n, k, k), A156991(n,k) = T(k, n, n).
Cf. A293616.

Programs

  • Maple
    A293617 := proc(m, n, k) option remember:
if m = 0 then 0^n elif k < 0 or k > n then 0 elif n = 0 then 1 else
(k+m)*A293617(m,n-1,k) + k*A293617(m,n-1,k-1) + A293617(m-1,n,k) fi end:
for m in [$0..4] do for n in [$0..6] do print(seq(A293617(m, n, k), k=0..n)) od od;
# Sample uses:
A027480 := n -> A293617(n, 2, 1): A293608 := n -> A293617(n, 4, 2):
# Flatten:
a := proc(n) local w; w := proc(k) local t, s; t := 1; s := 1;
while t <= k do s := s + 1; t := t + s od; [s - 1, s - t + k] end:
seq(A293617(n - k, w(k)[1], w(k)[2]), k=0..n) end: seq(a(n), n = 0..11);
  • Mathematica
    T[m_, n_, k_] := Pochhammer[m, k] StirlingS2[n + m, k + m];
For[m = 0, m < 7, m++, Print[Table[T[m, n, k], {n,0,6}, {k,0,n}]]]
A293617Row[m_, n_] := Table[T[m, n, k], {k,0,n}];
(* Sample use: *)
A293926Row[n_] := A293617Row[n, n];

Formula

T(m,n,k) = (k + m)*T(m, n-1, k) + k*T(m, n-1, k-1) + T(m-1, n, k) with boundary conditions T(0, n, k) = 0^n; T(m, n, k) = 0 if k<0 or k>n; and T(m, 0, k) = 0^k.
T(m,n,k) = Pochhammer(m, k)*binomial(n + m, k + m)*NorlundPolynomial(n - k, -k - m).
Showing 1-4 of 4 results.

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